It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the Riemann zeta function, verifies $\Lambda=\infty$. What would be the consequences on the distribution of prime numbers of such an equality?
Thank you in advance.

1 Answer
1

By itself, not very much: the assertion $\Lambda = \infty$ only requires the existence of an arbitrarily sparse sequence of pairs of adjacent zeroes whose normalised spacing goes to infinity arbitrarily slowly. Since the distribution of primes is controlled by the aggregate behaviour of the zeroes (through the explicit formula), such a sparse set of zeroes could have arbitrarily small impact on the distribution of the primes.

However, if one assumes stronger statements, then one can say something non-trivial about the primes. For instance, Goldston and Montgomery showed that Montgomery's pair correlation conjecture (which is (morally, at least) significantly stronger than $\Lambda = \infty$, as it controls the bulk distribution of gaps (or more precisely, differences between nearby zeroes), rather than the worst case distribution) is equivalent (on RH) to a variance asymptotic for the prime number theorem in short intervals. See also a later paper of Goldston showing a variant of this result. (Very roughly speaking, the explicit formula tells us that the variance asymptotic in the prime number theorem in short intervals is essentially a Fourier transform of the distribution of the differences between nearby zeroes of zeta.)

On the other hand, the negation $\Lambda < \infty$ of the assertion $\Lambda=\infty$ is a rather strong statement (indeed, likely too strong to be true, though we can't prove that yet), and should have noticeable consequences for the primes. For instance, it is plausible that one could show that $\Lambda < \infty$ is inconsistent with the pair correlation conjecture (certainly it is incompatible with the GUE hypothesis, which is stronger than pair correlation; also, the complementary assertion $\lambda>0$ to $\Lambda < \infty$ is certainly incompatible with pair correlation). If this was the case, one could take the contrapositive of the Goldston-Montgomery result, and infer that the expected variance asymptotic for the prime number theorem in short intervals cannot always hold, which in turn places limitations as to how strong and uniform an error term one can get in the Hardy-Littlewood prime tuples conjecture.

$\begingroup$Thank you very much for this wonderful answer. By the way, as English is not my mother tongue, can you tell me whether the spelling "zeros" is correct or not? I've had a doubt about it for quite a long time.$\endgroup$
– Sylvain JULIENOct 15 '14 at 16:47